An open connected set is called a domain " " : Carath�odory considers obviously disjoint sets.
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Such a path exists since " U " is assumed to be an open connected set.
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According to Hans Hahn, the concept of a domain as an open connected set was introduced by Constantin Carath�odory in his famous book.
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Here a major difference is evident from the one-variable theory : while for any open connected set in we can find a function that will nowhere continue analytically over the boundary, that cannot be said for.
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For example, in his influential monographs on elliptic partial differential equations, Carlo Miranda uses the term " region " to identify an open connected set, and reserves the term " domain " to identify an internally connected, perfect set, each point of which is an accumulation point of interior points, according to this convention, if a set is a region then its closure " } } is a domain.
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